I have always been bothered by the concept of “constructivism.“ which I understood as the idea that students construct their own understanding of math concepts, and a teacher’s carefully designed explanation can’t do that for them The problem with this is that while I believe it is true, it isn’t very helpful. How do they do that? How do I help them to do that, and help them to get better at it?

When I first tried to implement the idea 20+ years ago, I fell into a trap of thinking I just had to stop giving students the answers and they would start coming up with answers themselves. Sometimes people would say “students should discover things for themselves.” Any of you with me at the bottom of that pit know what a failure that is. And there is a much more subtle trap that I am still wrestling with - the idea that the less I say in math class the better, that I should always avoid just telling a student something. I suspect that we all wrestle with this every day, we know that we “shouldn’t” tell students what to do, but then there are those students who are struggling a bunch, so we give them a hint, or a suggestion, or an example, to help them get back in the game. But somehow that feels like cheating, or less than ideal.

There is often a vast gap between what most people imagine math to be (the process of remembering how to do something the teacher taught you, like adding fractions, and then getting the answer) and the math experience we are trying to build in our classrooms. The difference shows up in both the tasks we are asking the students to do, which is currently pretty well-defined and easy to observe, and the ways we hope students will engage with those tasks, which is less well-defined. In trying to better define that second piece I realized that the process I want my students to go through on a daily basis (almost minute by minute) is to build on what they know to figure out something they don’t yet know or understand. It is not so much about constructing understanding (which will happen, I assume, although I still don’t know how) as it is about asking students to construct their answers from the skills and knowledge they possess. Not “get” the answer through mental gifts or divine intervention, but through remembering, reasoning, modeling, etc., construct the answer.

Defining constructivism as “constructing answers” rather than “constructing understanding” is a subtle difference but it helps me a lot. “Constructing understanding” is almost a tautology – of course people construct their own understanding. It may be an insight, but it doesn’t really guide me in how to do things differently. “Constructing answers” reminds me that I am teaching not just individual skills, but also the process of selecting and sequencing those skills appropriately for a context. Students need opportunities to practice that before I tell them what to do. There is a criticism that problem-solving requires too much of novices, and should be the domain of experts. But I think it is a matter of scale. Why not include an appropriate, novice level of decision making in the work we give our students? Why not ask them what they notice or what they can figure out before we share our expert insights?

“Constructing answers” gives me insight about when it may be beneficial to step in or not step in, the pieces that may be missing for a given student or class, and the kinds of information I might share, questions to ask, etc. It is less about how much student talk vs. how much teacher talk, and more about who is talking when. I want my students to talk first, so I can see how they are constructing their answers. That way I will know what I have to say when it is my turn.

When I first tried to implement the idea 20+ years ago, I fell into a trap of thinking I just had to stop giving students the answers and they would start coming up with answers themselves. Sometimes people would say “students should discover things for themselves.” Any of you with me at the bottom of that pit know what a failure that is. And there is a much more subtle trap that I am still wrestling with - the idea that the less I say in math class the better, that I should always avoid just telling a student something. I suspect that we all wrestle with this every day, we know that we “shouldn’t” tell students what to do, but then there are those students who are struggling a bunch, so we give them a hint, or a suggestion, or an example, to help them get back in the game. But somehow that feels like cheating, or less than ideal.

There is often a vast gap between what most people imagine math to be (the process of remembering how to do something the teacher taught you, like adding fractions, and then getting the answer) and the math experience we are trying to build in our classrooms. The difference shows up in both the tasks we are asking the students to do, which is currently pretty well-defined and easy to observe, and the ways we hope students will engage with those tasks, which is less well-defined. In trying to better define that second piece I realized that the process I want my students to go through on a daily basis (almost minute by minute) is to build on what they know to figure out something they don’t yet know or understand. It is not so much about constructing understanding (which will happen, I assume, although I still don’t know how) as it is about asking students to construct their answers from the skills and knowledge they possess. Not “get” the answer through mental gifts or divine intervention, but through remembering, reasoning, modeling, etc., construct the answer.

Defining constructivism as “constructing answers” rather than “constructing understanding” is a subtle difference but it helps me a lot. “Constructing understanding” is almost a tautology – of course people construct their own understanding. It may be an insight, but it doesn’t really guide me in how to do things differently. “Constructing answers” reminds me that I am teaching not just individual skills, but also the process of selecting and sequencing those skills appropriately for a context. Students need opportunities to practice that before I tell them what to do. There is a criticism that problem-solving requires too much of novices, and should be the domain of experts. But I think it is a matter of scale. Why not include an appropriate, novice level of decision making in the work we give our students? Why not ask them what they notice or what they can figure out before we share our expert insights?

“Constructing answers” gives me insight about when it may be beneficial to step in or not step in, the pieces that may be missing for a given student or class, and the kinds of information I might share, questions to ask, etc. It is less about how much student talk vs. how much teacher talk, and more about who is talking when. I want my students to talk first, so I can see how they are constructing their answers. That way I will know what I have to say when it is my turn.